Laboratory Exercise #3 by S5k2Kcs

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									                  Beam Deflections and Material Properties Laboratory Exercise

Introduction
         Probably the most common type of structural member is the beam. A beam may be defined
as a member whose length is relatively large in comparison with its thickness and depth, and which
is loaded with transverse loads that produce significant bending effects as opposed to twisting or
axial effects. Whenever a real beam is loaded in any manner, the beam will deform such that an
initially straight beam will assume some deformed shape. Remember in Statics we assumed that a
beam was rigid, which is not the actual case. In ENGR350 Mechanics of Materials, many
relationships for beams will be developed. We will explore several of them. The first is:

                                              1 M
                                                                                            (1)
                                               EI

where  is the radius of curvature of the deflected beam at the point of interest. In addition, from
your statics background, M is the internal bending moment, E is Young’s modulus, and I is the area
moment of inertia of the cross sectional area of the beam about the neutral axis perpendicular to the
direction of the applied load on the beam. The area moment of inertia for a beam with a rectangular
cross section is given by the expression below:

                                                   1 3
                                             I      bh                                      (2)
                                                  12

where b is the width of the beam and h is the height or thickness of the beam.

Objectives
The objectives of this experiment are:
a) To explore beam deflections which result from externally applied loads. To accomplish this
     we will measure the Young's modulus of aluminum, brass, and steel.
b) To experimentally measure beam deflections and then compare them to analytical predictions.

Preparatory Exercise
In our January 23 lecture we conducted an experiment with a cantilever beam. The brass beam was
1.5 inches wide, 0.25 inches thick, and 14 inches long. Use data from lecture and equations (2) and
 (8) from this handout to calculate the Young’s modulus of the beam material. Compare the
calculated value to the tabulated value on page 54 of the “Structures” book.

Procedure
         This experiment involves using a simply supported beam loaded as shown in the next
figure. This beam is loaded in such a manner that the radius of curvature is constant between the
two supports. A mirror mounted to the beam changes angle as the beam is loaded. By reflecting a
laser beam off of the mirror and measuring the change in height of where the beam strikes a vertical
plane, we can determine the radius of curvature (). Knowing the moment (M) and the moment of
inertia (I) we can perform a calculation to find Young's Modulus (E).




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Part A (Determining Young’s Modulus)
        The following information should help you in understanding the fundamentals of beam
deflection and also allow you to make the proper calculations to determine Young’s Modulus for
the various beam materials.



                                      d


               e
                                                                          Mirror

                            Laser                                                 Beam
      Target                                  a              c            a

                                          P                                   P


For the above loading condition on the beam, the moment, M, is determined to be:

                                              M=Pa                                 (3)

From the following figure two more relationships can be found as shown below:
                          e/d = 2                               = c/2        (4)




                                               d


                      e
                                              2



                                                           c/2


                                                                 
                                                       
                                                           Angle of Incidence




                                                   2
If the last two relationships are combined the result is:

                                                = dc/e                                (5)

Putting all these relationships into the radius of curvature relationship leads to the following:

                                              1      Pa
                                                 =
                                              dc     b h3
                                                   E
                                              e       12

which can be rearranged to give the following:

                                                   Eb h3
                                              P=         e                              (6)
                                                   12dca

this has the form of y = mx+b where y = P and x = e. In Part A you will apply a load P and
measure an elevation e. You then will do a regression fit to find the slope m from which you can
calculate Young's Modulus E. The data that you will need to take is to be found in the Data Sheets.

Part B (Comparison of Measured and Predicted Beam Deflections)
        In this part you will take measurements of beam deflections and compare them to an
analytical prediction. The following equation applies to a cantilever beam as shown in the next
figure.

                                                P x2
                                           v=        (3L - x)                           (7)
                                                6EI

where (v) is the vertical deflection of the beam at any point along the length of the beam (x)




                                                   L


                                          x
                                                                         P




The Laboratory Assistant will demonstrate the experimental apparatus. See the Data Sheets for the
data that needs to be taken.




                                                   3
Part C (Deviation of Curvature Equation)
        One of the main assumptions related to beam deflections is that the deflections are small.
From this comes the simplification that when  is small, cos  1. However, when beams are
exposed to large deflections, this assumption breaks down. This part of the lab addresses the
breakdown of the “curvature” equation presented in Part A. For the cantilever beam similar to Part
B, the deflection equation for the “end” of the beam is

                                                 PL3
                                              v                                    (8)
                                                 3EI

Now apply loads in increments of 10 grams up to 100 grams and measure the deflections. See the
Data Sheets for the required data.

Laboratory Write-up
        Follow the laboratory report instructions previously provided. For Part A, do the values for
Young's modulus compare well with the values provided in Gordon on page 54? Also find two
other sources for Young's modulus. Are various sources consistent? How well should they
compare? You measured a, c, d, b, h, e, P during this experiment. An error in which of these
would cause the most error in the resulting calculation of Young's Modulus? Why? Draw the
general profile of the “Shear and Moment Diagram” for the loading conditions of Part A. Explain
why the radius of curvature is constant over the range of “c” from the figure on page 2.
        For Part B, prepare a plot of P (x-axis) versus v (y-axis) with the analytical expression
indicated as a solid line and the experimental data as points. How well do the two compare? What
is your definition of well in this case?
        For Part C, prepare a plot of P versus v with the analytical expression indicated as a solid
line and the experimental data as points. How well do the two compare? At what load does it
appear the experimental results start to diverge from the analytical expression? If the equation for
the slope of the beam at the point of loading is  = PL2/3EI, what is the angle  at the maximum
load applied? How about at the loading where the analytical and experimental results diverge?
What is the cos  value equal to at the angle you chose? Is it close to 1? Also, in looking at the
deflection of the beam at the higher loads, what geometrical changes are occurring to the loading
configuration that could influence the deflection results?




                                                 4
                                              Data Sheets
Part A:

Data you will need before beginning the experiment is as follows:

          Distance from target to mirror (d)=_____________

          Distance between supports (c)=________________

          Distance from load to support (a)=_____________

          Width of beam (b)=____________________

          Height of beam (h)=___________________

          Initial height mark of laser =___________________

The following table may be useful for taking data.

                                        Load P Versus Elevation e
             Aluminum                             Steel                         Brass
     Load            Elevation            Load            Elevation    Load         Elevation
      (lb)              (in)               (lb)              (in)       (lb)           (in)




Part B

Before starting this experiment you need to measure the following quantities:

          The beam width (b) =___________________

          The beam height (h) =__________________

          The beam length (L+h) =__________________

          The distance (x+h) to where



                                                   5
           you are measuring the deflection =____________________
           The initial micrometer reading (Dinitial) =_________________

The following table may be useful for data collection.

                                      Load P versus Deflection (v)
         Load            Dfinal             v*            Load             Dfinal            v*
          (lb)           (in)              (in)            (lb)            (in)             (in)




*
    v=Dinitial-Dfinal


Part C:

Before starting this experiment you need to measure the following quantities:

           The beam width (b) =___________________

           The beam height (h) =__________________

           The beam length (L+h) =

           The initial deflection of the beam due to its own weight (vinitial) =_________________




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